Optimal linear optical implementation of a single-qubit damping channel
Kent Fisher, Robert Prevedel, Rainer Kaltenbaek, Kevin J. Resch
OOptimal linear optical implementation of a single-qubit damping channel
Kent Fisher, ∗ Robert Prevedel, Rainer Kaltenbaek,
1, 2 and Kevin J. Resch Institute for Quantum Computing, Department of Physics and Astronomy,University of Waterloo, Waterloo, N2L 3G1, ON, Canada Vienna Center for Quantum Science and Technology, Faculty of Physics,University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria (Dated: October 1, 2018)We experimentally demonstrate a single-qubit decohering quantum channel using linear optics.We implement the channel, whose special cases include both the amplitude-damping channel and thebit-flip channel, using a single, static optical setup. Following a recent theoretical result [M. Piani et al. , Phys. Rev. A, , 032304 (2011)], we realize the channel in an optimal way, maximizingthe probability of success, i.e., the probability for the photonic qubit to remain in its encoding.Using a two-photon entangled resource, we characterize the channel using ancilla-assisted processtomography and find average process fidelities of 0 . ± . . ± . PACS numbers: 42.50.-p, 42.50.Ex, 03.67.-a, 03.65.Yz
Introduction.
Time evolution in quantum mechanicsconverts a density matrix to another density matrix. Thisevolution is referred to as a quantum channel and can bedescribed mathematically as a completely positive (CP)map [1]. Because of the generality of the concept of quan-tum channels, their use is ubiquitous in quantum infor-mation. For example, unitary quantum channels are usedin quantum computing to describe quantum gates. Non-unitary channels, on the other hand, describe the inter-action of quantum states with an environment, and haverecently been connected to fundamental physical ques-tions in quantum information science, such as channelcapacity, superadditivity [2, 3] and bound entanglement[4].Linear optics and single photons have several charac-teristics that make them an ideal testbed for quantuminformation. Single-qubit unitaries are easy to imple-ment as, for polarization encoded qubits, they only re-quire waveplates. Photonic qubits also exhibit long co-herence times, and spontaneous parametric down con-version allows the generation of high-quality entangledstates, which can be easily manipulated. However, cer-tain operations, such as the two-qubit CNOT-gate, aredifficult in this architecture [5, 6], and can only be im-plemented probabilistically [7–9].Unfortunately, the ease of single-qubit operations doesnot extend to more general CP maps. Some quantumchannels, like the depolarizing single-qubit channel [1]can be implemented with unit probability, but this is notthe case in general. For instance, the amplitude-dampingchannel, a non-unital quantum process, has been imple-mented in linear optics only with a limited success proba-bility of 1 / et al. [13] that, any single-qubit quantum channel could be implemented probabilisticallyusing linear optics and postselection, i.e., similar to manytwo-qubit operations. Moreover, they proved that suchimplementations can be achieved with the optimal suc-cess probability.In the present work, we use this recent theoretical re-sult to design and demonstrate a linear-optics-based im-plementation of a certain class of non-unital single-qubitquantum channels called “damping channels”. The classof channels we focus on can be parametrized by two realnumbers: α, β . In the operator-sum representation, thechannel’s action on an arbitrary quantum state ρ can bewritten as E ( ρ ) = (cid:80) i A i ρA † i , where the two Kraus oper-ators are [14]: A = (cid:18) cos α
00 cos β (cid:19) , A = (cid:18) β sin α (cid:19) (1)This channel is of great interest as its special cases in-clude the amplitude-damping ( α = 0) and bit-flip ( α = β ) channels, both of which are common sources of errorin other implementations of quantum information pro-cessing, such as ion traps. Furthermore, it belongs tothe small class of quantum channels for which the quan-tum capacity can be directly calculated via the coherentinformation [15].Here, we experimentally realize this single-qubit damp-ing quantum channel using linear optics. We can use thesetup to add controlled amounts of noise of various typesto a single qubit. The schematics of the experimentalsetup are shown in Fig. 1. The key step in the imple-mentation of the channel is the splitting of the polar-ization encoded information into different spatial modes,which then allows for the manipulation of different logicalstates independently of one another. An arrangement ofhalf-wave plates and liquid-crystal retarders allows us toprobabilistically implement both Kraus operators withina single, static optical setup. We characterize the chan-nel using a new ancilla-assisted quantum process tomog-raphy method, and show the optimality of our optical a r X i v : . [ qu a n t - ph ] S e p /4 Channel /4 Channel X // X a bc d B SourceSource A PBSQWPPBQQHWP
Beam displacer
Pol. control
LCR
Dichroic mirrorDetectoreDeFibre couplerP Fibr
FIG. 1: The experimental setup. We use spontaneousparametric down conversion in periodically poled KTiOPO (PPKTP) to generate entangled photon pairs of the form (cid:12)(cid:12) Φ + (cid:11) = √ ( | HH (cid:105) + | V V (cid:105) ) which are subsequently coupledinto single-mode fibres. One of the photons is sent throughthe damping channel parameterized by α and β , which are setby the angles a , b , c and d of four half-wave plates (HWPs).Two liquid-crystal retarders (LCRs) switch anti-correlativelybetween the identity, 1 , and the Pauli X operation. The po-larization of each photon is measured by an analyzer (A andB) consisting of a half- and a quarter-wave plate (QWPs) fol-lowed by a polarizing beam splitter (PBS). Eventually, thephotons are detected by single-photon counting modules. implementation, with our success rates in the amplitude-damping case surpassing those of previous implementa-tions [10, 11]. In order to characterize the action of thechannel on entanglement we study the amount of entan-glement of photon pairs when one photon is sent throughthe channel. Optimality of the implementation.
Following Ref. [13],it can be shown that the probability of success fora specific Kraus decomposition { A i } is p succ ( { A i } ) = (cid:16)(cid:80) i (cid:107) A i (cid:107) ∞ (cid:17) − , where the norm (cid:107) M (cid:107) ∞ is the largestsingular value of the operator M . Maximizing over allpossible Kraus decompositions A i describing the chan-nel allows to achieve the optimal success probability p optsucc = max A i (cid:80) i (cid:107) A i (cid:107) ∞ . For our particular channel, ifwe assume that cos( α ) ≥ cos( β ), this yields: p optsucc = 1cos α + sin β (2)In order to achieve p optsucc , we have to implement eachKraus operator with individual probabilities p A i = (cid:107) A i (cid:107) ∞ · p optsucc . We find that the optimal probability ofsuccess is achieved for p A = cos α cos α +sin β and p A = sin β cos α +sin β . We show experimentally that for variousvalues of α and β , which can be independently controlledin our experiment, we indeed achieve this upper bound. Ancilla-assisted quantum process tomography.
Quan-tum process tomography (QPT) allows to experimentallyreconstruct the superoperator describing an unknownphysical process. Ancilla-assisted QPT (AAQPT) uses ancillary qubits to facilitate the reconstruction procedurefor quantum state measurements. It has also been shown[16] that AAQPT gives decreasing statistical errors as theamount of entanglement between the primary and ancillasystems is increased.AAQPT has been used to study various unitary quan-tum gates [16] but has not yet been extended to the char-acterization of non-unital channels. In our work, and incontrast to previous AAQPT schemes [17–19], we do notassume an ideal description of our initial state, but rathermeasure and include it when using a maximum-likelihoodtechnique to find the physical matrix that best describesthe action of the experimentally implemented channel.The standard techniques for QPT and AAQPT aredescribed in [1] and [16], respectively. Below, we out-line our method following their nomenclature. Considera two-qubit state, ρ AB , whose density matrix is known;e.g., it might have been reconstructed using quantumstate tomography (QST). The quantum channel E actson subsystem A, while subsystem B is unaffected. Thetransformed two-qubit state after the channel is ρ (cid:48) AB =( E ⊗
1) ( ρ AB ). Characterizing ρ (cid:48) AB , e.g., by performingstandard QST, allows for reconstruction of the processusing the Choi–Jamio(cid:32)lkowski isomorphism [20, 21].The quantum process can then be written as ρ (cid:48) AB = (cid:80) d − m,n =0 χ mn ( ˜ E m ⊗ ρ AB ( ˜ E n ⊗ † where { ˜ E i } are op-erators which form a basis in the space of d × d matrices( d = 2 in our case). It is common to use the basis formedby the Pauli matrices { , X, Y, Z } . The d -dimensionalprocess matrix χ then fully describes the quantum pro-cess. In our maximum-likelihood technique, we parame-terize χ by d − d = 16 real numbers [18, 19] and seekto minimize the following function: f = ν (cid:88) i =1 ( n i − N Tr[ M i ρ (cid:48) AB ]) N Tr[ M i ρ (cid:48) AB ] + λ (cid:88) k (cid:34)(cid:88) m,n χ mn Tr( ˜ E † n ˜ E m ˜ E k ) − Tr( ˜ E k ) (cid:35) , (3)such that the resulting χ most closely resembles a phys-ical quantum process. Here, i labels the measurementsetting in the final QST, ν is the number measurementsettings, n i is the number of two-fold coincidence countsrecorded in the i th setting, N corresponds to the numberof photons incident on the detectors, M i is the projectorof the i th measurement, and λ is a Lagrange multiplierused to force the resulting process matrix to be tracepreserving [17]. Experiment.
We use the experimental setup shown inFig. 1 to implement the quantum channel defined by theKraus operators in Eq. 1. Two 40 mm calcite beam dis-placers are used to construct an interferometer. Withinthese beam displacers, photons with horizontal ( | H (cid:105) ) andvertical ( | V (cid:105) ) polarization are spatially displaced with re-spect to each other [9]. Half-wave plates (HWPs) areused to set the amount of damping by allowing to adjust α and β in Eq. 1. The relations between these parame- ou t FIG. 2: (a) Probability of success as a function of dampingparameter β for cases α = 0, α = β and α = β . The shadedregion below 0.5 represents probabilities of success in previousoptical implementations of the amplitude-damping channel,see Refs. [10–12] (b) The tangle, τ , of the resulting two-photonstate as a function of damping β after one photon has passedthrough the damping channel. Errors in the experimentaldata are calculated from Poissonian noise in the coincidencecounts and are not visible on the scale of the plots. Thesolid lines in both panels represent the theoretically expecteddependance. In (b), the experimental density matrix of theinput state was used for the calculations. The shaded regionsaround the theory curves in both plots represent the expectedstandard deviation assuming 1 ◦ and 1% rotation errors in theHWPs and LCRs respectively. The margin of error in the α = β case is significantly smaller than for the other casesdue to the fact that the HWP angles and LCR settings alllie at points where partial derivatives of the Kraus-operatormatrix elements are zero. ters and the individual angles a, b, c, d of the four HWPsare given by sin 4 a = cos β cos α , b = a − π , sin 4 c = − sin α sin β and d = π − c . The channel is realized by switchingrandomly between the Kraus operators A and A . Theswitching is performed using two liquid-crystal retarders(LCRs). We set the two LCRs to X and , respec-tively, to implement A , and we set them to and X to implement A . Here, the subscripts represent the ac-tion of the first and the second LCR. The probabilities, p A and p A , with which each configuration is realizedare determined by the values of α and β such that theoverall success probability of realizing the channel is op- timal [13]. The switching rate of the LCRs was chosento be 10 Hz, significantly faster than the integration timefor a single measurement (5 s).To characterize the channel, we use the AAQPTscheme outlined above. Our resource state is an entan-gled photon pair generated in a type-II down conversionsource in a Sagnac configuration [22, 23]. A 0.5 mW laserdiode at 404.5 nm pumps a 25 mm periodically-poledcrystal of KTiOPO (PPKTP). This typically yielded acoincidence rate of 10 kHz. The characterization of thechannel is executed as follows: The HWP angles a , b , c and d are set to zero and the LCRs to X and suchthat the channel acts as the identity map. A QST is per-formed to obtain the density matrix of the input state, ρ AB . The HWP angles and the probabilities for switch-ing the LCRs and the HWP angles are then set accord-ing to the values of α and β . Another QST yields theoutput state, ρ (cid:48) AB . QST involves recording coincidencesfor all combinations of the eigenstates of the Pauli X, Yand Z operators. For each of these 36 projective mea-surements, we integrated coincidence counts for 5 s. Theresulting data were then used in conjunction with Eq. 3to reconstruct the superoperator describing the quantumprocess. Results.
We now turn to our main result, the opti-mality of our quantum channel implementation. Fig. 2ashows the probability of success for the amplitude-damping, bit-flip, and one in-between case ( α = β ).Since amplitude-damping manifests itself as photon lossin our particular implementation, determining the prob-ability of success reduces to measuring the transmissionof the channel. The experimental data closely followthe theoretical predictions (solid lines) that are basedon Eq. 2.Previous optical implementations of the amplitude-damping channel [10, 11] have given at most 50% prob-ability of success [24], whereas here we find that only inthe case of maximum damping ( β = π/
2) the probabilityof success decreases to 50%. The experimental resultsfor the success probability closely resemble the theoret-ical prediction. This is also true for our experimentalimplementation of the bit-flip channel and the α = β case of single-qubit damping.Fig. 2b shows the tangle [25] of the two-photon out-put density matrix, ρ (cid:48) AB , for the amplitude-damping,bit-flip, and α = β cases, where one of the two pho-tons passes through the quantum channel. Theoreticalcurves are based on the action of the respective idealquantum channels on the experimental input state. Itcan be seen in each case, that the experimental dataagrees well with the theoretical prediction, showing thatthe experimentally implemented channel closely resem-bles the ideal one. The process fidelity is defined by F =Tr (cid:112) √ χ exp χ id √ χ exp [26], where χ exp and χ id are the ex-perimental and ideal process matrices, respectively. Wefind process fidelities of 0 . ± . . ± . . ± . α = 0, α = β and α = β casesof the channel, respectively. We also compute the so- Im( exp )Re( exp ) Im( id )Re( id ) (b)(a) P r o c e ss F i de li t y T r a c e D i s t an c e FIG. 3: (a) Measured process fidelity (solid data points) andtrace distance (unfilled data points) as a function of dampingfor each of the three cases studied. Error bars ( ∼ − ),calculated using Monte-Carlo simulations adding Poissoniannoise to the measured state tomography counts in each run,are too small to see on this scale. (b) Real and imaginary partsof the experimentally determined and ideal process matricesat maximum amplitude-damping ( α = 0 , β = π/ called maximum trace distance [1], which is defined as D = max ρ in Tr (cid:12)(cid:12) ρ expout − ρ idout (cid:12)(cid:12) , where | A | = √ A † A and ρ exp/idout = (cid:80) m,n χ exp/id mn ˜ E m ρ in ˜ E † n . Operationally, D cor-responds to the highest probability of distinguishing be-tween the experimental and ideal channels using the bestpossible input state. The average maximum trace dis-tance over all damping values measured were found tobe 0 . ± . . ± . . ± . χ matrix for α = 0, β = π/ Summary.
Decoherence plays an important role inquantum information science. Investigating its effectsrequires careful and well-controlled implementations ofthese noisy processes. Non-unital damping channels, likethe ones studied here, are crucial in further understand-ing quantum communication, in determining channel ca-pacities and for the generation of bound-entangled states.We have implemented a general damping single-qubit quantum channel with linear optics in which both typeand amount of decohering noise can be precisely con-trolled. A single, static optical setup can perform as theamplitude-damping channel, the bit-flip channel, or moregeneral cases characterized by two real parameters, α and β . Most importantly, we have shown that this channelhas been implemented in an optimal way, so as to maxi-mize the probability of success. The channels were char-acterized using a new approach to ancilla-assisted processtomography and, in all cases, operate with high fidelity. Acknowledgements.
We acknowledge fruitful discus-sions with J. Lavoie, M. Piani and N. L¨utkenhaus, andare grateful for financial support from Ontario Min-istry of Research and Innovation ERA, QuantumWorks,NSERC, OCE, Industry Canada and CFI. R.P. acknowl-edges support by the Ontario MRI and the Austrian Sci-ence Fund (FWF). ∗ Electronic address: k8fi[email protected][1] M.A. Nielsen and I.L. Chuang,
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